On the hessian of the optimal transport potential

Stefán Ingi Valdimarsson

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 3, page 441-456
  • ISSN: 0391-173X

Abstract

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We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.

How to cite

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Valdimarsson, Stefán Ingi. "On the hessian of the optimal transport potential." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.3 (2007): 441-456. <http://eudml.org/doc/272250>.

@article{Valdimarsson2007,
abstract = {We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.},
author = {Valdimarsson, Stefán Ingi},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Monge-Kantorovich mass transport problem; bounds for the Hessian of the optimal transport potential; Brascamp-Lieb inequalities},
language = {eng},
number = {3},
pages = {441-456},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the hessian of the optimal transport potential},
url = {http://eudml.org/doc/272250},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Valdimarsson, Stefán Ingi
TI - On the hessian of the optimal transport potential
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 3
SP - 441
EP - 456
AB - We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.
LA - eng
KW - Monge-Kantorovich mass transport problem; bounds for the Hessian of the optimal transport potential; Brascamp-Lieb inequalities
UR - http://eudml.org/doc/272250
ER -

References

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  1. [1] S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in n , Geom. Dedicata74 (1999), 201–212. Zbl0927.52007MR1674116
  2. [2] F. Barthe, On a reverse form of the Brascamp–Lieb inequality, Invent. Math.134 (1998), 335–361. Zbl0901.26010MR1650312
  3. [3] J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., to appear. Zbl1132.26006MR2377493
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  5. [5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math.44 (1991), 375–417. Zbl0738.46011MR1100809
  6. [6] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys.214 (2000), 547–563. Zbl0978.60107MR1800860
  7. [7] L. A. Caffarelli, Erratum: Monotonicity of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), 547–563; Comm. Math. Phys. 225 (2002), 449–450. Zbl0978.60107
  8. [8] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math.177 (1996), 113–161. Zbl0887.49017MR1440931
  9. [9] R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, Cambridge, 1985. Zbl1267.15001MR832183
  10. [10] E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math.102 (1990), 179–208. Zbl0726.42005MR1069246
  11. [11] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. Zbl1106.90001MR1964483
  12. [12] C. Villani, “Optimal Transport, Old and New”, 2007, July 18, preprint. Zbl1156.53003MR2459454

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